{"paper":{"title":"An asymptotic formula for Goldbach's conjecture with monic polynomials in $\\mathbb{Z}[\\theta][x]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ab\\'ilio Lemos, Anderson L. A. Araujo","submitted_at":"2013-12-27T16:45:19Z","abstract_excerpt":"In this paper, we consider $D=\\mathbb{Z}[\\theta]$, where\n  $$\\theta= \\sqrt{-k} \\,\\,\\,\\, \\mbox{if}\\;\\;\\;-k\\not\\equiv 1 \\;(\\mbox{mod}\\;4)\\,\\,\\,\\,\\mbox{or}\\,\\,\\,\\, \\theta=\\frac{\\sqrt{-k}+1}{2} \\,\\,\\,\\, \\mbox{if}\\;\\;\\;-k\\equiv 1 \\;(\\mbox{mod}\\;4),$$\n  $k\\geq 2$ is a squarefree integer, and we proved that the number $R(y)$ of representations of a monic polynomial $f(x)\\in \\mathbb{Z}[\\theta][x]$, of degree $d\\geq 1$, as a sum of two monic irreducible polynomials $g(x)$ and $h(x)$ in $\\mathbb{Z}[\\theta][x]$, with the coefficients of $g(x)$ and $h(x)$ bounded in complex modulus by $y$, is asymptotic t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.7295","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}